Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

1,891 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
9
votes
0answers
289 views

Validity of confidence interval for $\rho$ when $X\sim N_3(0,\Sigma)$ with $\Sigma_{ij}=\rho^{|i-j|}$

Suppose $X\sim N_3(0,\Sigma)$, where $\Sigma=\begin{pmatrix}1&\rho&\rho^2\\\rho&1&\rho\\\rho^2&\rho&1\end{pmatrix}$. On the basis of one observation $x=(x_1,x_2,x_3)'$, I ...
9
votes
0answers
301 views

MLE, regularity conditions, finite and infinite parameter spaces

The problem I have is in figuring out why the MLE is no longer consistent in countable parameter spaces under conditions specified below. The set up is as follows: we are consider a parameters space ...
7
votes
0answers
132 views

Understanding equation used by Hastie et al

I am trying to recreate FIGURE 3.6 from Elements of Statistical Learning. The only information about the figure is included in the caption. I am not clear on what the equation on the Y-axis means ...
7
votes
0answers
219 views

Rao-Blackwellization in variational inference

The Black box VI paper introduces Rao-Blackwellization as a method to reduce the variance of the gradient estimator using score function, in section 3.1. However I don't quite get the basic idea ...
7
votes
1answer
281 views

Sufficient statistics for $\mu_1 - \mu_2$

If $ X_1, ..., X_n$ is a random sample from $ X \sim N(\mu_1, \sigma^2)$ and $Y_1,..., Y_n$ is a random sample from $Y \sim N(\mu_2, \sigma^2),$ if the samples are independent and $ \sigma^2$ is known,...
6
votes
1answer
157 views

Help understanding a paragraph in Kadane's book Principles of uncertainty

Consider two infinite sequences of indicators of events, $s_1$ and $s_2$, with respective relative frequencies $l_1$ and $l_2$, where $l_1\neq l_2$. Let $A$ be the indicator of an event not an ...
6
votes
0answers
840 views

Exercise on Borel Cantelli Lemma ($\limsup X_n/ \ln(n) =1$ a.s.) help required to rigorously write the statement

I hope this question is within the scope of this site. Please note that I have solved this Exercise, I do have doubts about my presentation though and about how to rigorously empathize on the ...
6
votes
0answers
1k views

Dealing with auxiliary random variables for Mean-Field Variational Inference in Bayesian Poisson factorization

I am studying as a part of a class assignment a recent paper on Poisson factorization. Some points of the paper regarding the usage of some auxiliary variables are not clear to me. I would like to ...
6
votes
0answers
128 views

Question 10.9 from Bayesian Data Analysis, what does accuracy mean here?

I'm doing an independent study in Bayesian Statistics following some chapters from BDA3. When solving the first question from Ch 10 I got stuck. It says: [If] a scalar variable $\theta$ is ...
5
votes
0answers
47 views

How to show a UMVUE exists only if $g(p)$ is a polynomial of degree at most $n$?

Let $X\sim Bin(n,p)$. The problem is to show that a UMVUE can exist for $g(p)$ only if $g(p)$ is a polynomial in $p$ of degree at most $n$. For the case when $g(p) = \frac{1}{p}$ we can show that it ...
5
votes
0answers
204 views

Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
5
votes
0answers
86 views

French website Providing Instruction/Tutorials on Statistical Theory

This is somewhat of an odd question for CV, but since it's a question about statistical education, I think it falls within the scope of CV. Several years ago I stumbled across a French website that ...
5
votes
1answer
77 views

Functions of continuous random variables

Let Y be an exponential random variable with parameter $\tau > 0$. Compute the cdf and pdf of $F_W$ where $W = Y^3$ The solution states the cdf as $1 - e^{\frac{-y^\frac{1}{3}}{t}}$ because $F_Y =...
5
votes
0answers
233 views

I'm not asking for a conjugate prior. Is there a distribution $p(x|y)$ that satisfies $\int p(x|y)Beta(y|a,b) dy = Beta(x| a', b')$?

I know the result of integrating a Gaussian against another Gaussian is still Gaussian, $$\int N(x|\mu_y,\sigma_y)N(y|\mu,\sigma) dy = N(x|\mu',\sigma')\quad.$$ Can I get the same form for Beta ...
5
votes
0answers
586 views

Equality vs. Equality in Distribution ($t$-distribution for example)

A technical question that came up to mind as I was reading up on linear models today. Consider the $t$-distribution with $\nu$ degrees of freedom ($t_\nu$) for example. Let's say $T \sim t_{\nu}$; ...
5
votes
0answers
234 views

Combination of letters: some repeated letters

So I was looking for an answer on this assignment we have to program but I cannot find it anywhere. I'm a computer science student, not a statistics students. (And it isn't even for a statistics ...
5
votes
0answers
169 views

Simple $\chi^2$ test question

I have the following question which seems extremely easy, but the way the data are set up is causing me some uncertainty: I plan to solve this problem through finding the maximum likelihood estimate ...
5
votes
0answers
164 views

Exponential family where set of natural parameters has empty interior

In my math-stat class we have a theorem that goes: Let $\{P_\theta : \theta \in \Theta\}$ be a $k$ parameter exponential family (i.e. the density of a member of this family can be written as $f(\...
5
votes
0answers
310 views

Deriving priors for MCMC implementation

I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution. The posterior distribution is generated from a ...
5
votes
0answers
4k views

Conjugate of Weibull with shape known

This isn't exactly a homework problem but rather a self-selected problem I'm doing to prepare for a midterm. I can see from Wikipedia that it is an inverse gamma but I am unable to reach the ...
4
votes
0answers
61 views

Range of values of $R^2$ for a two-feature linear model based on the $R^2$s of one-feature linear models?

I was asked this in an interview. You have two features, $x_1$ and $x_2$. You fit a simple linear model on each feature, so $$ \underbrace{y = x_1 \beta}_{\text{model 1}}, \qquad \underbrace{y = x_2 \...
4
votes
0answers
87 views

Deriving spectral measure

While reading this book, I got stuck on page 266 where the authors found the spectral measure $F(du)$ of the generalized covariance function $K(h) = \Gamma(-\alpha/2) |h|^{\alpha}, ~0<\alpha<2.$ ...
4
votes
0answers
31 views

How is EMSE derived for causal trees in Athey and Imbens (PNAS 2016)?

Athey and Imbens build a non-parametric matching procedure to identify and estimate causal effects. To this end, they minimize the expected mean squared error (EMSE) of their procedure, but I don't ...
4
votes
0answers
45 views

Figuring out the margin for the soft margin SVM (exam question)

This is an exam question and I am not sure whether it is solveable with the given information. We were given a graphic that displayed binary labelled points $x^{(i)}\in \mathbb{R}^2$ with $y^{(i)} \...
4
votes
0answers
35 views

Integration by sampling from truncated distribution

I'm reading the book Ben Lambert's Bayesian Statistics: problems and answers, which by the way I like. There is a group of problems in "Integration by Sampling" chapter 12. The first integral is $$...
4
votes
0answers
161 views

Test for Lipschitz continuity (is there some?)

Let $x_1, \dots, x_n$ be a random sample from a distribution $D$. Say, I want to test whether $F(z)$, the cdf of $D$, is Lipschitz continuous, i.e. there exists $L$ such that $F(z + \delta) - F(z) \...
4
votes
0answers
120 views

Minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed

What is the minimum-variance unbiased estimator to estimate quantiles when the errors are normal distributed? median When we wish to estimate the median, $\mu$, of a normal distributed variable then ...
4
votes
0answers
38 views

Uniform distribution on the simplex. - Thomas cover

I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...
4
votes
0answers
806 views

Fisher information matrix in logistic regression

I am self-studying the basics of logistic regression. I came across this sentence: In logistic regression expected and observed information matrixes are equal I am aware that the information ...
4
votes
1answer
91 views

Convergence in Distribution for i.i.d. data

Let $X_1,X_2,\ldots,X_n$ be i.i.d. RVs with $E(X_{i})=\mu$ and $V(X_{i})=\sigma^2$, $\sigma <\infty$.Is it possible to find real sequences $a_{n}$ and $b_{n}$ such that $a_{n}(\bar{X}^3_{n}-b_{n})$ ...
4
votes
0answers
2k views

Generate Beta distribution from Uniform random variables

I need to generate random numbers from Beta distribution using random variables from Uniform distribution. If I have two random variables $Y_1=U_1^{1/\alpha}$ and $Y_2=U_1^{1/\beta}$, and If $Y_1+Y_2&...
4
votes
1answer
185 views

What is the likelihood function of this random variable (beta distribution parameterizing a Bernoulli distribution)?

This is related to an earlier self-study question of mine. The setup is that there are $N$ individuals, indexed by $i$, and two time periods. Individuals choose whether to "invent" something in the ...
4
votes
0answers
636 views

Concentration of maximum of subexponential random variables

I'm looking for a concentration bound on the maximum of a collection of sub-exponential random variables, which are not necessarily independent. More specifically, I have the following collection: \...
4
votes
0answers
497 views

Back-forecasting in MA(2) model

The sales of a certain product are represented by the model $$Z_t=3+a_t+0.5a_{t-1}-0.25a_{t-2}$$ where $a_t\sim WN(0,4)$ (White Noise). Given the data $Z_1=3.25,Z_2=4.75,Z_3=2.25$ and $Z_4=...
4
votes
0answers
37 views

Can someone clearly paraphrase the following argument?

Metaculus is a site where users make and justify predictions on various questions. My question is about an estimation of the probability that a human will live to 120 years by the year 2024. I am ...
4
votes
0answers
137 views

Does Cross-Validation Really Work?

This question was taken from "The Elements of Statistical Learning" by Friedman, Hastie and Tibshirani (question 7.10) Consider a scenario with N = 20 samples in two equal-sized classes, and p = 500 ...
4
votes
1answer
518 views

Model selection and estimation for pseudo out-of-sample forecasting

I have quarterly data on inflation from 1990 Quartal 1 to 2016 Quartal 3. If I want to perform the pseudo out-of-sample forecasting one quarter ahead with an autoregressive function, do I have to ...
4
votes
0answers
276 views

Describe AR process with additive white noise using ARMA process

Disclaimer: This is a homework problem This is a problem from "Adaptive Filter Theory" by Haykin. Problem 2.10 (2nd edition). Problem A discrete-time stochastic process $\{x(n)\}$ that is real-...
4
votes
1answer
109 views

Bayes net probability question

I've made this Bayes net based on a problem and I'm trying to find the probability of W but I'm stuck. I know I probably have to use Bayes theorem backwards through to find $P(W)$, but I'm not sure ...
4
votes
0answers
468 views

Interpretation of Kaplan-Meier Curve that doesn't go to 0

I'm making a survival analysis and I founded a little strange the survival curve founded by the Kaplan-Meier estimator. The curves are below Here each colour represents a group and time is in days. ...
4
votes
0answers
151 views

Comparison of Difference of Expectations of Conditional Variances

I want to show (if possible) that $$\mathrm{E}[\mathrm{Var(Y|X_1, X_2)}] - \mathrm{E}[\mathrm{Var(Y|X_1)}] \geq \mathrm{E}[\mathrm{Var(Y|X_1, X_2, X_3)}] - \mathrm{E}[\mathrm{Var(Y|X_1, X_3)}] \tag 1$...
4
votes
2answers
61 views

What statistical test do I need?

Say I have $N$ light bulbs. Whevener one breaks down, I immediately fix it. $k_0$ of these $N$ light bulbs do not break down during this year, $k_1$ break down (and get fixed) once, and $k_2$ ...
4
votes
1answer
613 views

Likelihood of LDA compared to logistic regression

I've come across an interesting exercise. We are given four classification models for binary response and a $d$-dimensional independent variable: A Linear Discriminant Analysis model where the ...
4
votes
0answers
91 views

Relating sufficient statistics to parameters

I'm studying sufficient statistics and I came across this problem: A dataset consists of independent triples $(W_i,Y_i,Z_i)$ of independent random variables with distributions as follows, $$ W_i \sim ...
4
votes
0answers
74 views

Problem involving P.D.F. containing an indicator variable

Let $X_1, X_2, \ldots$ be independently and identically distributed random variables with probability density functions: $$f(x) = \alpha \;x^{-(\alpha+1)} \; I_{(x>1)}, \; \; \alpha > 0.$$ For ...
4
votes
0answers
215 views

Think Bayes - Chapter 7 Exercice 7.4

I'm reading this book by Allen B. Downey and trying to do the exercises http://greenteapress.com/wp/think-bayes/ I am a bit stuck at this one, 7.4. I tried looking for blogs and stuff like that where ...
4
votes
0answers
105 views

what is this property? $\int p(x,\pi)d\pi=p(x|E[\pi])$?

Sorry if the title does not make sense, from the answer of this question Mistake in derivation about categorical distribution and Dirichlet distribution? it can be shown that say $p(x|\pi)$ follows ...
4
votes
0answers
219 views

Sign and size of OLS bias for Tobit models

I have a question related to the sign and size of the OLS bias in the case of a Tobit model. Consider the following model (1) Sample of observations $\{X_i,Y_i\}_{i=1}^n$, i.i.d., $X_i$ is a vector ...
4
votes
2answers
59 views

Time series and images : difference and terminology

A time series is an ordered collection of random variables. Considering a one-dimensional time series $A_i = {a_{i1},a_{i2},\ldots,a_{it}}$ where $t$ denotes the time index. So, the time series is a ...
4
votes
0answers
371 views

Finding the uniformly most powerful test

Let $X_1,X_2,...,X_n$ denote a random sample from density, $$f(x;\theta)={1\over 2\theta}, \quad 0<x<2\theta.$$ Find the uniformly most powerful test for testing $H_0:\theta \le \theta_0$ vs ...

1
2 3 4 5
38